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Modeling acid dissociation constant of analytes in binary solvents at various temperatures using Jouyban–Acree model
Thermochimica Acta 428 (2005) 119–123
Modeling acid dissociation constant of analytes in binary solvents at various temperatures using Jouyban–Acree model Abolghasem Jouybana,∗ , Somaieh Soltanib , Hak-Kim Chanc , William E. Acree Jr.d a
School of Pharmacy and Drug Applied Research Center, Tabriz University of Medical Sciences, 29th Bahman St., Tabriz 51664, Iran b Kimia Research Institute, P.O. Box 51665-171, Tabriz, Iran c Faculty of Pharmacy, The University of Sydney, Sydney, NSW 2006, Australia d Department of Chemistry, University of North Texas, Denton 76203-5070, USA Received 18 August 2004; received in revised form 19 October 2004; accepted 1 November 2004 Available online 13 December 2004
Abstract A mathematical equation, namely Jouyban–Acree model, for calculating apparent acid dissociation constants (pKa ) in hydro-organic mixtures with respect to the concentration of organic solvent and temperature is proposed. The correlation ability of the model is evaluated by employing pKa values of 17 different acids in water–cosolvent systems. The results show that the model is able to correlate the pKa values with an overall average percentage differences (APD) of 1.71 ± 1.86%. In order to test the prediction capability of the model, nine experimental pKa values from each data set have been employed to train the model, then the pKa values at other solvent compositions and temperatures were predicted and the overall APD obtained is 2.10 ± 2.42%. The applicability of the extended form of Jouyban–Acree model on pKa data of analytes in ternary solvent mixtures is also shown. © 2004 Elsevier B.V. All rights reserved. Keywords: Cosolvency; Acid dissociation constant; Hydro-organic mixtures; Different temperatures; Jouyban–Acree model
1. Introduction During the lead optimization phase of drug discovery studies, where a large number of molecules are biologically evaluated in parallel, the determination of physico-chemical properties is essential to ensure adequate characterization and quality of developed candidates [1]. The knowledge of acid dissociation constants of these new chemical entities is of fundamental importance in order to provide information for scientists working on chromatographic separations (retention times and selectivity dependence of mobile phase pH) [2], capillary electrophoresis separations (migration times or mobilities of the ionic species over a range of pH values) [2], pharmaceutical drug discovery and development [3], chemical reactivity (selection of conditions for synthesis by considering the effects of pH on reaction products and proper∗
Corresponding author. Tel.: +98 411 3392584; fax: +98 411 3344798. E-mail address: [email protected] (A. Jouyban).
0040-6031/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.tca.2004.11.002
ties of postulated intermediates), salt formation, purification process [1] and pharmaceutical formulation development [1]. Experimental measurements of pKa values are expensive, difficult, time consuming and limited by purity of compounds (almost all experimental methods except capillary electrophoresis) [3], low analyte solubility (in potentiometry), the range of pH (in high performance liquid chromatography), spectral similarities (in spectrometric methods), and stability of analytes (e.g. chemical reactions intermediates). The most important limitation is that before synthesis of a compound, its pKa value cannot be estimated experimentally. Although water is the most common solvent in chemical/pharmaceutical applications, organic solvents are used as a cosolvent in order to adjust separation selectivity and modify solubility, stability, pKa and other characteristics of the analytes. Temperature plays a significant role in most of the chromatographic and electrophoretic methods and it is recognized as the most relevant parameter in gas chromatography. Many separation scientists have traditionally
120
A. Jouyban et al. / Thermochimica Acta 428 (2005) 119–123
disregarded temperature effects in liquid chromatography whereas elevated temperature is a usual controlling variable in reversed phase liquid chromatography. Today publications stated the reasons for fearing of such methods and now temperature in combination with pH and cosolvent addition are introduced as variables to adjust selectivity towards acidic and basic compounds e.g. see Refs. [4–6]. Since the pKa value of many compounds is determined commonly at 25 ◦ C, prediction of pKa at different temperatures (e.g. body temperature 37 ◦ C) would be very useful tool for biological and biomedical applications. Using mixed solvents in the analytical/pharmaceutical areas is a common method to optimize solubility and/or separation efficiency. However, the number of solvent compositions and temperature combinations is quite large and it is difficult to determine all possible combinations by experiments. Thus, a good alternative is to use computational methods. The aim of this work is to present a mathematical treatment of pKa values as a function of solvent composition and temperature. The accuracy of the proposed model is assessed by using available pKa values in mixed solvents at various temperatures collected from the literature.
2. Theoretical treatment
Summation of Eqs. (2)–(4) yields: + µm − µm µm HA H+ A− w w = Xc (µcH+ + µcA− − µcHA ) + Xw (µw H+ + µA− − µHA )
+ (A0 + B0 − C0 )Xc Xw + (A1 + B1 − C1 )Xc Xw (Xc − Xw )
(5)
Replacing the corresponding equals from Eq. (1) into Eq. (5) with appropriate rearrangements give: 2.303R log Kam,T = Xc (2.303R log Kac,T ) + Xw (2.303R log Kaw,T ) Xc Xw T c X Xw (Xc − Xw ) + (A1 + B1 − C1 ) T + (A0 + B0 − C0 )
(6)
Since (A0 + B0 − C0 ), (A1 + B1 − C1 ) and 2.303R are constant values and pKa = −log Ka , it is possible to simplify Eq. (6) as: pKam,T = Xc pKac,T + Xw pKaw,T + W0 +W1
Xc Xw T
Xc Xw (Xc − Xw ) T
(7)
A dissociation reaction of a monoprotic acid (HA) in a solvent can be represented as: a + · a A− HA ↔ H+ + A− , Ka = H aHA
where W0 = (A0 + B0 − C0 )/2.303R and W1 = (A1 + B1 − C1 )/ 2.303R. It is obvious that one can use the volume /weight fractions of the solvents instead of the mole fractions [7] and rewrite Eq. (7) as:
where a is the activity of the chemical species. The logarithm of Ka expressed as:
pKam,T = f c pKac,T + f w pKaw,T + K0
2.303RT log Ka = µH+ + µA− − µHA
(1)
where µ denotes chemical potential of the species. The chemical potential in aqueous cosolvent mixtures can be expressed as the mole fractions of the cosolvents: µm = Xc µcH+ + Xw µw + A0 X c X w H+ H+ + A1 Xc Xw (Xc − Xw )
(2)
µm = Xc µcA− + Xw µw + B0 X c X w A− A− + B1 Xc Xw (Xc − Xw )
(3)
c c w w c w µm HA = X µHA + X µHA + C0 X X
+ C1 Xc Xw (Xc − Xw )
+ K1
f c f w (f c − f w ) T
where superscripts m, c and w denote mixed solvent, pure cosolvent and pure water, respectively, X is the mole fraction of the solvents, and A, B and C the solute–solvent and solvent–solvent interaction terms. These terms represent the two body and three body interactions in the solution [7].
(8)
where K0 and K1 are the curve-fitting parameters. The numerical values of K0 and K1 can be computed by fitting the experimental values of (pKam,T − f c pKac,T − f w pKaw,T ) against c w c w f cf w and f f (fT −f ) by using a no intercept least square T analysis. The general form of the proposed equation can be expressed as: pKam,T = f c pKac,T + f w pKaw,T + f cf w
(4)
f cf w T
n Kq (f c − f w )q q=0
T
(9)
The model could possess as many curve-fitting parameters as needed for accurate representation of experimental data. However, it is preferred to employ the lowest number of curve-fitting parameters, since it requires minimum number of experimental data in model training process. In some cases, the numerical values of pKac are not available. If so, it is pos-
A. Jouyban et al. / Thermochimica Acta 428 (2005) 119–123
sible to rewrite Eq. (8) as: pKam,T = f w pKaw,T + Jf c + K0 + K1
3. Results and discussion f cf w T
f c f w (f c − f w ) T
(10)
where J, K0 and K1 are the model constant. These are computing via fitting (pKam,T − f w pKaw,T ) against fc , f c f w and f c f w (f c − f w ). Eq. (10) can be written as Eq. (11) for calculating pKa values in mixed solvents at a fixed temperature [7]: pKam,T = f w pKaw + Mf c + M0 f c f w +M1 f c f w (f c −f w ) (11) where M, M0 and M1 are the model constants. Eq. (11) produced reasonably accurate results for both calculation of pKa of a single analyte and a number of related analytes in binary solvents [7]. Eq. (11) can also be extended to calculate pKa of analytes in ternary solvent mixtures as: pKam = f w pKaw + f c pKac + f s pKas + Q0 f w f c +Q1 f w f c (f w − f c ) + Q 0 f w f s +Q 1 f w f s (f w − f s ) + Q 0 f c f s w c s +Q 1 f c f s (f c − f s ) + Q 0f f f w c s w c s +Q 1 f f f (f − f − f )
(12)
where superscript s denote the second cosolvent, and Q terms are the model constants. If the numerical values of pKa of analytes in pure solvents (e.g. pKac or pKas ) are not available, it is possible to consider them as constant terms. The other versions of Eq. (9) have been successfully applied for calculation of solute solubilities in mixed solvents at various temperatures [8], dielectric constants [9] and also surface tensions [10] of liquid mixtures. The model could be considered empirical in nature, however, it produces accurate calculations and has its position in data modeling of physico-chemical properties in mixed solvents. To assess the accuracy of the proposed equations, the average percentage differences (APD) between experimental and calculated pKa values are considered as an accuracy criterion: APD =
121
In order to evaluate the accuracy of the proposed model, available pKa values in different concentrations of the organic solvents at various temperatures including more than 15 data points were collected from the literature. The details of the collected data including the solute, the cosolvent, the number of experimental data points in each set, the temperature range and the references are listed in Table 1. To evaluate the correlation ability of the Jouyban–Acree model, all data points in each set have been fitted to Eq. (9) with q = 0, 1 and 2, and the back calculated pKa values have been used to compute APDs. This numerical method has been called correlative analysis and the results are shown in Table 1. The overall APD and the standard deviations for q = 0, 1, and 2 are 1.71 ± 1.86, 1.26 ± 1.30 and 0.99 ± 0.85, respectively. There is no significant differences in the overall APD between q = 0 and q = 1 or between q = 1 and 2 (paired t-test, P > 0.07). Therefore, Eq. (9) with q = 0 is selected as appropriate version of the model. For Eq. (9) possessing three constant terms (i.e. pKaw,T , J and K0 ), the least APD value (0.27%) has been observed for eriochrome black T in waterethanol and the highest APD value (4.98%) has been observed for maleic acid in water–ethanol mixtures. The IPD values produced by correlative analysis of Eq. (11) with q = 0 sorted in three subgroups, i.e. IPD ≤ 2, 2–4 and >4% are illustrated in Fig. 1. As seen in more than 75% of the cases the IPD is less than 2% whereas the experimental relative standard deviation for repeated experiments is up to 20% [3]. In order to evaluate the prediction capability of the selected version of the Jouyban–Acree model, nine experimental pKa values (high, low and medium T, f c and f w ) have been used to train the model and then the trained models are then employed to predict the pKa values of remainder points in each data set. The minimum prediction APD (0.19%) is observed for adipic acid in water–methanol and the maximum value (9.12%) for pKa2 of tartaric acid in water–ethanol mixtures. The overall APD obtained is 2.10 ± 2.42%. Fig. 1 also shows the relative frequency of IPD values for predictive analysis. As expected, error percentage less than 2% shows the highest relative frequency.
100 pKacalculated − pKaobserved N pKaobserved
where N denotes the number of experimental data point in each set. The individual percentage difference (IPD) is calculated by: pKcalculated − pKobserved a a IPD = 100 pKaobserved
Fig. 1. Relative frequency of IPD values for correlative and predictive analyses.
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A. Jouyban et al. / Thermochimica Acta 428 (2005) 119–123
Table 1 Details of the systems studied and the average percentage deviation (APD) for correlative and predictive analyses No.
Solute
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Adipic acid, pKa1 Adipic acid, pKa2 EBBc , pKa1 EBBc , pKa2 EBBc , pKa1 EBBc , pKa2 EBTd , pKa1 EBTd , pKa2 Maleic acid, pKa1 Maleic acid, pKa2 Phthalic acid, pKa1 Phthalic acid, pKa2 PANe Tartric acid, pKa1 Tartric acid, pKa2 Terazodone Trisf
Cosolvent
Methanol Methanol Dimethylformamide Dimethylformamide Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol 2-Methoxyethanol
Temperature (◦ C)
30–60 30–60 20–40 20–40 20–45 20–45 20–40 20–40 30–55 30–55 30–55 30–55 20–30 30–55 30–55 15–45 20–45
Reference
[14] [14] [15] [15] [15] [15] [15] [15] [16] [16] [16] [16] [17] [16] [16] [18] [19]
Na
16 16 26 26 30 30 25 25 36 36 36 36 33 36 36 36 50
Overall APD S.D. a b c d e f
Correlative
Predictive, q = 0b
q=0
q=1
q=2
0.35 2.07 0.55 0.48 0.53 0.44 0.27 0.47 4.98 1.91 1.55 1.24 4.94 1.68 6.21 0.71 0.68
0.35 2.07 0.49 0.44 0.53 0.43 0.22 0.42 4.77 1.70 1.40 1.15 1.37 1.06 4.03 0.7 0.32
0.35 2.07 0.42 0.41 0.52 0.40 0.22 0.39 3.05 1.62 1.30 1.08 1.15 0.55 2.46 0.70 0.17
0.53 2.53 0.64 0.57 0.63 0.49 0.3 0.49 5.63 2.22 2.56 1.37 5.12 1.99 9.12 0.66 1.08
1.71 ± 1.86
1.26 ± 1.30
0.99 ± 0.85
2.10 ± 2.42
The number of data points in each set. APD of predictive analysis using models trained by nine experimental data points. The number of predicted points is N − 9. Eriochrome blue black RC. Eriochrome black T. 1-(2-Pyridylazo)-2-naphthol. Tris-(hydroxymethyl) aminomethane.
Four available pKa data sets in ternary solvents at 25 ◦ C taken from the literature [11] are used to check the applicability of Eq. (12) to reproduce such data. The APDs of pKa of heptanoic, hexanoic, pentanoic and butanoic acids in water–methanol–dioxane are 0.41, 0.38, 0.48 and 0.39%, respectively and the overall APD is 0.42%. All data points of alkanoic acids in water–methanol–dioxane are fitted to Eq. (12) and the obtained APD is 0.56%. To check the prediction capability of the model to predict the pKa data, experimental data of heptanoic and butanoic acids are used to train the model, and then pKa of hexanoic and pentanoic acids are predicted. The resulted prediction of APD is 0.51%, which shows a good predictability of the proposed model. It is obvious that such data is required in analytical [12] and pharmaceutical [13] areas where ternary solvents are used for mobile phases and/or drug formulations. To the best of our knowledge, there is no published pKa data in ternary solvents at various temperatures to evaluate the accuracy of the proposed model. However, it is anticipated that the model is able to predict such data and it could be employed in generating pKa data in ternary and even higher order multicomponent systems at various temperatures after training by a minimum number of experimental data. In conclusion, the APD value for correlative studies using 17 data sets and employing a three constant term model is 1.71 ± 1.86% (N = 529) and for predicted data points using the trained models the APD value is 2.10 ± 2.42% (N = 376). These low correlation and prediction errors mean that the
proposed model is able to calculate pKa values in binary solvents at various temperatures within an acceptable error range. The corresponding values for pKa of analytes in ternary solvents at a fixed temperature are 0.56 and 0.51%, respectively. It is suggested that employing the proposed models could help the researchers to speed up the optimization procedure.
References [1] G.A. Caliaro, C.A. Herbotes, J. Pharm. Biomed. Anal. 26 (2001) 427. [2] G. Lunn, HPLC Methods for Pharmaceutical Analysis, Wiley, Germany, 2000. [3] A. Avdeef, K.J. Box, J.E.A. Comer, M. Gilges, M. Hadley, C. Hibbert, W. Patterson, K.Y. Tam, J. Pharm. Biomed. Anal. 20 (1999) 631. [4] Y. Guillaume, C. Guinchard, J. Liq. Chromatogr. 17 (1994) 2809. [5] C.B. Castells, C. Rafols, M. Roses, E. Bosch, J. Chromatogr. A 1002 (2003) 41. [6] S. Heinisch, J.L. Rocca, J. Chromatogr. A 1048 (2004) 183. [7] A. Jouyban, H.K. Chan, B.J. Clark, W.E. Acree Jr., Int. J. Pharm. 246 (2002) 135. [8] A. Jouyban-Gh., W.E. Acree Jr., Int. J. Pharm. 167 (1998) 177. [9] A. Jouyban, Sh. Soltanpour, H.-K. Chan, Int. J. Pharm. 269 (2004) 353. [10] A. Jouyban, A. Fathi-Azarbayjani, W.E. Acree Jr., Chem. Pharm. Bull. 52 (2004) 1219. [11] G. Papanastasiou, I. Ziogas, I. Moumtzis, Anal. Chim. Acta 186 (1986) 213.
A. Jouyban et al. / Thermochimica Acta 428 (2005) 119–123 [12] A. Jouyban, S.C. Grosse, H.-K. Chan, M.W. Coleman, B.J. Clark, J. Chromatogr. A 994 (2003) 191. [13] A. Martin, P.L. Wu, T. Velasquez, J. Pharm. Sci. 74 (1985) 277. [14] G.A. El-Naggar, Talanta 47 (1998) 1013. [15] M.S. Masoud, H.H. Hammud, H. Beidas, Thermochim. Acta 381 (2002) 119.
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[16] A.A. Zaghloul, G.A. El-Naggar, Sh.A. El-Shazly, M.F. Amira, Talanta 45 (1997) 189. [17] L. Coo, J.T. Cardwell, R.W. Cattrall, S.D. Kolev, Anal. Chim. Acta 360 (1998) 153. [18] I. Shehatta, Helv. Chim. Acta 85 (2002) 2125. [19] A.A. El-Harakany, F.M. Abdel Halim, A.O. Barakat, J. Electroanal. Chem. 162 (1984) 285.
Modeling acid dissociation constant of analytes in binary solvents at various temperatures using Jouyban–Acree model Abolghasem Jouybana,∗ , Somaieh Soltanib , Hak-Kim Chanc , William E. Acree Jr.d a
School of Pharmacy and Drug Applied Research Center, Tabriz University of Medical Sciences, 29th Bahman St., Tabriz 51664, Iran b Kimia Research Institute, P.O. Box 51665-171, Tabriz, Iran c Faculty of Pharmacy, The University of Sydney, Sydney, NSW 2006, Australia d Department of Chemistry, University of North Texas, Denton 76203-5070, USA Received 18 August 2004; received in revised form 19 October 2004; accepted 1 November 2004 Available online 13 December 2004
Abstract A mathematical equation, namely Jouyban–Acree model, for calculating apparent acid dissociation constants (pKa ) in hydro-organic mixtures with respect to the concentration of organic solvent and temperature is proposed. The correlation ability of the model is evaluated by employing pKa values of 17 different acids in water–cosolvent systems. The results show that the model is able to correlate the pKa values with an overall average percentage differences (APD) of 1.71 ± 1.86%. In order to test the prediction capability of the model, nine experimental pKa values from each data set have been employed to train the model, then the pKa values at other solvent compositions and temperatures were predicted and the overall APD obtained is 2.10 ± 2.42%. The applicability of the extended form of Jouyban–Acree model on pKa data of analytes in ternary solvent mixtures is also shown. © 2004 Elsevier B.V. All rights reserved. Keywords: Cosolvency; Acid dissociation constant; Hydro-organic mixtures; Different temperatures; Jouyban–Acree model
1. Introduction During the lead optimization phase of drug discovery studies, where a large number of molecules are biologically evaluated in parallel, the determination of physico-chemical properties is essential to ensure adequate characterization and quality of developed candidates [1]. The knowledge of acid dissociation constants of these new chemical entities is of fundamental importance in order to provide information for scientists working on chromatographic separations (retention times and selectivity dependence of mobile phase pH) [2], capillary electrophoresis separations (migration times or mobilities of the ionic species over a range of pH values) [2], pharmaceutical drug discovery and development [3], chemical reactivity (selection of conditions for synthesis by considering the effects of pH on reaction products and proper∗
Corresponding author. Tel.: +98 411 3392584; fax: +98 411 3344798. E-mail address: [email protected] (A. Jouyban).
0040-6031/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.tca.2004.11.002
ties of postulated intermediates), salt formation, purification process [1] and pharmaceutical formulation development [1]. Experimental measurements of pKa values are expensive, difficult, time consuming and limited by purity of compounds (almost all experimental methods except capillary electrophoresis) [3], low analyte solubility (in potentiometry), the range of pH (in high performance liquid chromatography), spectral similarities (in spectrometric methods), and stability of analytes (e.g. chemical reactions intermediates). The most important limitation is that before synthesis of a compound, its pKa value cannot be estimated experimentally. Although water is the most common solvent in chemical/pharmaceutical applications, organic solvents are used as a cosolvent in order to adjust separation selectivity and modify solubility, stability, pKa and other characteristics of the analytes. Temperature plays a significant role in most of the chromatographic and electrophoretic methods and it is recognized as the most relevant parameter in gas chromatography. Many separation scientists have traditionally
120
A. Jouyban et al. / Thermochimica Acta 428 (2005) 119–123
disregarded temperature effects in liquid chromatography whereas elevated temperature is a usual controlling variable in reversed phase liquid chromatography. Today publications stated the reasons for fearing of such methods and now temperature in combination with pH and cosolvent addition are introduced as variables to adjust selectivity towards acidic and basic compounds e.g. see Refs. [4–6]. Since the pKa value of many compounds is determined commonly at 25 ◦ C, prediction of pKa at different temperatures (e.g. body temperature 37 ◦ C) would be very useful tool for biological and biomedical applications. Using mixed solvents in the analytical/pharmaceutical areas is a common method to optimize solubility and/or separation efficiency. However, the number of solvent compositions and temperature combinations is quite large and it is difficult to determine all possible combinations by experiments. Thus, a good alternative is to use computational methods. The aim of this work is to present a mathematical treatment of pKa values as a function of solvent composition and temperature. The accuracy of the proposed model is assessed by using available pKa values in mixed solvents at various temperatures collected from the literature.
2. Theoretical treatment
Summation of Eqs. (2)–(4) yields: + µm − µm µm HA H+ A− w w = Xc (µcH+ + µcA− − µcHA ) + Xw (µw H+ + µA− − µHA )
+ (A0 + B0 − C0 )Xc Xw + (A1 + B1 − C1 )Xc Xw (Xc − Xw )
(5)
Replacing the corresponding equals from Eq. (1) into Eq. (5) with appropriate rearrangements give: 2.303R log Kam,T = Xc (2.303R log Kac,T ) + Xw (2.303R log Kaw,T ) Xc Xw T c X Xw (Xc − Xw ) + (A1 + B1 − C1 ) T + (A0 + B0 − C0 )
(6)
Since (A0 + B0 − C0 ), (A1 + B1 − C1 ) and 2.303R are constant values and pKa = −log Ka , it is possible to simplify Eq. (6) as: pKam,T = Xc pKac,T + Xw pKaw,T + W0 +W1
Xc Xw T
Xc Xw (Xc − Xw ) T
(7)
A dissociation reaction of a monoprotic acid (HA) in a solvent can be represented as: a + · a A− HA ↔ H+ + A− , Ka = H aHA
where W0 = (A0 + B0 − C0 )/2.303R and W1 = (A1 + B1 − C1 )/ 2.303R. It is obvious that one can use the volume /weight fractions of the solvents instead of the mole fractions [7] and rewrite Eq. (7) as:
where a is the activity of the chemical species. The logarithm of Ka expressed as:
pKam,T = f c pKac,T + f w pKaw,T + K0
2.303RT log Ka = µH+ + µA− − µHA
(1)
where µ denotes chemical potential of the species. The chemical potential in aqueous cosolvent mixtures can be expressed as the mole fractions of the cosolvents: µm = Xc µcH+ + Xw µw + A0 X c X w H+ H+ + A1 Xc Xw (Xc − Xw )
(2)
µm = Xc µcA− + Xw µw + B0 X c X w A− A− + B1 Xc Xw (Xc − Xw )
(3)
c c w w c w µm HA = X µHA + X µHA + C0 X X
+ C1 Xc Xw (Xc − Xw )
+ K1
f c f w (f c − f w ) T
where superscripts m, c and w denote mixed solvent, pure cosolvent and pure water, respectively, X is the mole fraction of the solvents, and A, B and C the solute–solvent and solvent–solvent interaction terms. These terms represent the two body and three body interactions in the solution [7].
(8)
where K0 and K1 are the curve-fitting parameters. The numerical values of K0 and K1 can be computed by fitting the experimental values of (pKam,T − f c pKac,T − f w pKaw,T ) against c w c w f cf w and f f (fT −f ) by using a no intercept least square T analysis. The general form of the proposed equation can be expressed as: pKam,T = f c pKac,T + f w pKaw,T + f cf w
(4)
f cf w T
n Kq (f c − f w )q q=0
T
(9)
The model could possess as many curve-fitting parameters as needed for accurate representation of experimental data. However, it is preferred to employ the lowest number of curve-fitting parameters, since it requires minimum number of experimental data in model training process. In some cases, the numerical values of pKac are not available. If so, it is pos-
A. Jouyban et al. / Thermochimica Acta 428 (2005) 119–123
sible to rewrite Eq. (8) as: pKam,T = f w pKaw,T + Jf c + K0 + K1
3. Results and discussion f cf w T
f c f w (f c − f w ) T
(10)
where J, K0 and K1 are the model constant. These are computing via fitting (pKam,T − f w pKaw,T ) against fc , f c f w and f c f w (f c − f w ). Eq. (10) can be written as Eq. (11) for calculating pKa values in mixed solvents at a fixed temperature [7]: pKam,T = f w pKaw + Mf c + M0 f c f w +M1 f c f w (f c −f w ) (11) where M, M0 and M1 are the model constants. Eq. (11) produced reasonably accurate results for both calculation of pKa of a single analyte and a number of related analytes in binary solvents [7]. Eq. (11) can also be extended to calculate pKa of analytes in ternary solvent mixtures as: pKam = f w pKaw + f c pKac + f s pKas + Q0 f w f c +Q1 f w f c (f w − f c ) + Q 0 f w f s +Q 1 f w f s (f w − f s ) + Q 0 f c f s w c s +Q 1 f c f s (f c − f s ) + Q 0f f f w c s w c s +Q 1 f f f (f − f − f )
(12)
where superscript s denote the second cosolvent, and Q terms are the model constants. If the numerical values of pKa of analytes in pure solvents (e.g. pKac or pKas ) are not available, it is possible to consider them as constant terms. The other versions of Eq. (9) have been successfully applied for calculation of solute solubilities in mixed solvents at various temperatures [8], dielectric constants [9] and also surface tensions [10] of liquid mixtures. The model could be considered empirical in nature, however, it produces accurate calculations and has its position in data modeling of physico-chemical properties in mixed solvents. To assess the accuracy of the proposed equations, the average percentage differences (APD) between experimental and calculated pKa values are considered as an accuracy criterion: APD =
121
In order to evaluate the accuracy of the proposed model, available pKa values in different concentrations of the organic solvents at various temperatures including more than 15 data points were collected from the literature. The details of the collected data including the solute, the cosolvent, the number of experimental data points in each set, the temperature range and the references are listed in Table 1. To evaluate the correlation ability of the Jouyban–Acree model, all data points in each set have been fitted to Eq. (9) with q = 0, 1 and 2, and the back calculated pKa values have been used to compute APDs. This numerical method has been called correlative analysis and the results are shown in Table 1. The overall APD and the standard deviations for q = 0, 1, and 2 are 1.71 ± 1.86, 1.26 ± 1.30 and 0.99 ± 0.85, respectively. There is no significant differences in the overall APD between q = 0 and q = 1 or between q = 1 and 2 (paired t-test, P > 0.07). Therefore, Eq. (9) with q = 0 is selected as appropriate version of the model. For Eq. (9) possessing three constant terms (i.e. pKaw,T , J and K0 ), the least APD value (0.27%) has been observed for eriochrome black T in waterethanol and the highest APD value (4.98%) has been observed for maleic acid in water–ethanol mixtures. The IPD values produced by correlative analysis of Eq. (11) with q = 0 sorted in three subgroups, i.e. IPD ≤ 2, 2–4 and >4% are illustrated in Fig. 1. As seen in more than 75% of the cases the IPD is less than 2% whereas the experimental relative standard deviation for repeated experiments is up to 20% [3]. In order to evaluate the prediction capability of the selected version of the Jouyban–Acree model, nine experimental pKa values (high, low and medium T, f c and f w ) have been used to train the model and then the trained models are then employed to predict the pKa values of remainder points in each data set. The minimum prediction APD (0.19%) is observed for adipic acid in water–methanol and the maximum value (9.12%) for pKa2 of tartaric acid in water–ethanol mixtures. The overall APD obtained is 2.10 ± 2.42%. Fig. 1 also shows the relative frequency of IPD values for predictive analysis. As expected, error percentage less than 2% shows the highest relative frequency.
100 pKacalculated − pKaobserved N pKaobserved
where N denotes the number of experimental data point in each set. The individual percentage difference (IPD) is calculated by: pKcalculated − pKobserved a a IPD = 100 pKaobserved
Fig. 1. Relative frequency of IPD values for correlative and predictive analyses.
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Table 1 Details of the systems studied and the average percentage deviation (APD) for correlative and predictive analyses No.
Solute
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Adipic acid, pKa1 Adipic acid, pKa2 EBBc , pKa1 EBBc , pKa2 EBBc , pKa1 EBBc , pKa2 EBTd , pKa1 EBTd , pKa2 Maleic acid, pKa1 Maleic acid, pKa2 Phthalic acid, pKa1 Phthalic acid, pKa2 PANe Tartric acid, pKa1 Tartric acid, pKa2 Terazodone Trisf
Cosolvent
Methanol Methanol Dimethylformamide Dimethylformamide Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol Ethanol 2-Methoxyethanol
Temperature (◦ C)
30–60 30–60 20–40 20–40 20–45 20–45 20–40 20–40 30–55 30–55 30–55 30–55 20–30 30–55 30–55 15–45 20–45
Reference
[14] [14] [15] [15] [15] [15] [15] [15] [16] [16] [16] [16] [17] [16] [16] [18] [19]
Na
16 16 26 26 30 30 25 25 36 36 36 36 33 36 36 36 50
Overall APD S.D. a b c d e f
Correlative
Predictive, q = 0b
q=0
q=1
q=2
0.35 2.07 0.55 0.48 0.53 0.44 0.27 0.47 4.98 1.91 1.55 1.24 4.94 1.68 6.21 0.71 0.68
0.35 2.07 0.49 0.44 0.53 0.43 0.22 0.42 4.77 1.70 1.40 1.15 1.37 1.06 4.03 0.7 0.32
0.35 2.07 0.42 0.41 0.52 0.40 0.22 0.39 3.05 1.62 1.30 1.08 1.15 0.55 2.46 0.70 0.17
0.53 2.53 0.64 0.57 0.63 0.49 0.3 0.49 5.63 2.22 2.56 1.37 5.12 1.99 9.12 0.66 1.08
1.71 ± 1.86
1.26 ± 1.30
0.99 ± 0.85
2.10 ± 2.42
The number of data points in each set. APD of predictive analysis using models trained by nine experimental data points. The number of predicted points is N − 9. Eriochrome blue black RC. Eriochrome black T. 1-(2-Pyridylazo)-2-naphthol. Tris-(hydroxymethyl) aminomethane.
Four available pKa data sets in ternary solvents at 25 ◦ C taken from the literature [11] are used to check the applicability of Eq. (12) to reproduce such data. The APDs of pKa of heptanoic, hexanoic, pentanoic and butanoic acids in water–methanol–dioxane are 0.41, 0.38, 0.48 and 0.39%, respectively and the overall APD is 0.42%. All data points of alkanoic acids in water–methanol–dioxane are fitted to Eq. (12) and the obtained APD is 0.56%. To check the prediction capability of the model to predict the pKa data, experimental data of heptanoic and butanoic acids are used to train the model, and then pKa of hexanoic and pentanoic acids are predicted. The resulted prediction of APD is 0.51%, which shows a good predictability of the proposed model. It is obvious that such data is required in analytical [12] and pharmaceutical [13] areas where ternary solvents are used for mobile phases and/or drug formulations. To the best of our knowledge, there is no published pKa data in ternary solvents at various temperatures to evaluate the accuracy of the proposed model. However, it is anticipated that the model is able to predict such data and it could be employed in generating pKa data in ternary and even higher order multicomponent systems at various temperatures after training by a minimum number of experimental data. In conclusion, the APD value for correlative studies using 17 data sets and employing a three constant term model is 1.71 ± 1.86% (N = 529) and for predicted data points using the trained models the APD value is 2.10 ± 2.42% (N = 376). These low correlation and prediction errors mean that the
proposed model is able to calculate pKa values in binary solvents at various temperatures within an acceptable error range. The corresponding values for pKa of analytes in ternary solvents at a fixed temperature are 0.56 and 0.51%, respectively. It is suggested that employing the proposed models could help the researchers to speed up the optimization procedure.
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